Numerical Simulation of Hydraulic Characteristics in a Vortex Drop Shaft

water

Article Numerical Simulation of Hydraulic Characteristics in A Vortex Drop Shaft

Wenchuan Zhang , Junxing Wang *, Chuangbing Zhou, Zongshi Dong and Zhao Zhou State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, China; [email protected] (W.Z.); [email protected] (C.Z.); [email protected] (Z.D); [email protected] (Z.Z.) * Correspondence: [email protected]; Tel.: +86-137-0718-2138

Received: 18 July 2018; Accepted: 25 September 2018; Published: 28 September 2018

Abstract: A new type of vortex drop shaft without ventilation holes is proposed to resolve the problems associated with insufficient aeration, negative pressure (Unless otherwise specified, the pressure in this text is gauge pressure and time-averaged pressure) on the shaft wall and cavitation erosion. The height of the intake tunnel is adjusted to facilitate aeration and convert the water in the intake tunnel to a non-pressurized flow. The hydraulic characteristics, including the velocity (Unless otherwise specified, the velocity in this text is time-averaged velocity), pressure and aeration concentration, are investigated through model experiment and numerical simulation. The results revealed that the RNG k-ε turbulence model can effectively simulate the flow characteristics of the vortex drop shaft. By changing the inflow conditions, water flowed into the vertical shaft through the intake tunnel with a large amount of air to form a stable mixing cavity. Frictional shearing along the vertical shaft wall and the collisions of rotating water molecules caused the turbulence of the flow to increase; the aeration concentration was sufficient, and the energy dissipation effect was excellent. The cavitation number indicated that the possibility of cavitation erosion was small. The results of this study provide a reference for the analysis of similar spillways.

Keywords: vortex drop shaft; turbulence model; spillway aerators; energy dissipation; cavitation

1. Introduction Compared with traditional energy dissipators, vortex drop shaft spillways can rapidly change the flow regime and form areas of turbulence or whirlpools to dissipate energy. These spillways provide various functions, such as flood discharge control, energy dissipation and drainage structure protection. Additionally, they transfer the energy dissipation task from outside to inside to avoid outlet atomization (in the traditional discharge energy dissipation process, water interacts with the air boundary to form an atomized flow [1]). However, within these spillways, the pressure near the vertical shaft wall gradually decreases because of gravity and wall friction, and generated negative pressure can easily cause cavitation erosion [2,3]. Therefore, considerable attention has been paid to negative pressure and cavitation erosion in the applications of vortex drop shaft spillways. To solve these problems, erosion reduction devices are often installed in vortex drop shaft spillways to ensure the stable operation of the discharge structures. Many studies have investigated the characteristics of vortex drop shaft spillways, mainly focusing on the diameter of the vertical shaft and the depth of the dissipation well; they found that the energy dissipation effect could be improved by optimizing the vertical shaft shape [4]. Design schemes of vortex chambers and vertical shafts were proposed based on previous research on vortex drop shaft spillways, and the design criteria was summarized [5,6]. Based on an analysis of the influence of the body shape on the hydraulic parameters of an energy dissipation well, one study found that a

Water 2018, 10, 1393; doi:10.3390/w10101393 www.mdpi.com/journal/water Water 2018, 10, 1393 2 of 18 reasonable depth for an energy dissipation well should be 1.69 times the vertical shaft diameter [7]. Other studies focused on the tangential slot vortex intakes of vertical shafts for urban drainage, and they found that the flow in a tapering and downward-sloping vortex inlet channel was strongly dependent on the geometry of the inlet and vertical shaft and that the hydraulic instability was related to the discharge [8,9]. Del Giudice and Gisonni [10] found that an appropriate length of the lowered bottom could prevent free surface fluctuations at the intake structure, resulting in an appropriate vortex air core without choking problems for supercritical approach flows. In addition, the hydraulic characteristics of swirling flows and the energy dissipation rate were both related to the swirl number. The larger the swirl number was, the greater the energy dissipation. Furthermore, the tangential velocity ensured no negative pressure on the vertical shaft wall and enhanced the flow stability [11]. The radial distribution of the pressure in the swirl zone complied with the theoretical distribution of the pressure in combined eddies, and the pressure increased as the radius increased [12]. In previous studies, the velocity of a cavity swirl was assumed based on the theory of combined vortex and free vortex, and a corresponding pressure formula was obtained [13,14]. In another investigation, the need for aeration facilities was emphasized, and a method for calculating the aeration cavity length of annular aeration in a swirling vertical shaft was derived based on projectile theory [15]. Dong et al. [16] used the standard k-ε turbulence model to simulate a vortex drop shaft spillway and obtained the hydraulic parameters of spiral flow. Gao et al. [17] simulated the characteristics of turbulent flow through a vertical pipe inlet/outlet with a horizontal anti-vortex plate. However, compared with the standard k-ε turbulence model, the RNG k-ε turbulence model can more effectively process flows with high strain rates and streamline bending by correcting the turbulent viscosity and considering both rotation and swirling in the average flow regime. In addition, a previous study found that the water surface was more stable using this approach, and the simulation results agreed well with experimental data [18,19]. For traditional vortex drop shaft spillways, the water in the intake tunnel moves with pressure flow, and thus ventilation holes are set in the volute chamber or in front of the outlet tunnel [2,5,20]. Moreover, an annular aerator was set in the middle of the shaft to increase the aeration concentration and avoid erosion [15]. In practical engineering, aeration facilities often cause great difficulties during the process of construction. Therefore, to resolve the insufficient energy dissipation, cavitation of traditional spillways and construction difficulty, a vortex drop shaft with an increased-height intake tunnel and without ventilation holes is investigated in the present study. The pressure field, flow field and energy dissipation are simulated and analyzed by the renormalization group (RNG) k-ε turbulence model [21], and the results are compared with experimental data to better understand the distribution trends of the hydraulic parameters, the characteristics of the energy dissipation and the cavitation associated with a vortex drop shaft.

2. Physical Model A Froude similitude with a geometric scale ratio of 1:25 was applied in the physical model. The model scale has a slight effect on the time-averaged hydraulic characteristics such as the time-averaged water depth, pressure and velocity, and the scale effects could be negligible, so they can provide useful references for practical engineering [22]. Small-scale models based upon the Froude similitude may underestimate the air transport in the fluid because the relative effects of the surface tension and viscosity are over-represented, especially when the scale is smaller than 1:30 [23–26]. In summary, a model experiment can simulate the time-averaged water depth, pressure and velocity well, and the aeration concentration is underestimated so that it is larger in the prototype. A higher aeration concentration is more favourable for preventing cavitation corrosion, so the experimental results can be used as a reliable reference for engineering design. The vortex drop shaft investigated in this study consists of a tangential intake tunnel, a volute chamber, a gradient section, a vertical shaft, a dissipation well, and an outlet tunnel. The intake tunnel has an arched shape, with a slope and length of 1:7.5 and 0.68 m, respectively, and the section size Water 2018, 10, 1393 3 of 18 changed from 0.2 m × 0.24 m (width × height) to 0.176 m × 0.33 m (width × height). The diameter of the voluteWater 2018 chamber, 10, x FOR is PEER 0.352 REVIEW m, and the height from the gradient section to the top of volute chamber3 of 19 is 0.72 m. The diameter and depth of the vertical shaft are 0.216 m and 1.82 m, respectively. The depth of the dissipationchanged from well 0.2 ism 0.28 × 0.24 m m and (width the height× height) of to gradient 0.176 m × section 0.33 m (width is 0.24 × m. height). The outlet The diameter tunnel of has an archedthe shape, volute chamber with a width, is 0.352 height, m, and the length height and from slope the ofgradient 0.2 m, section 0.24 m, to the 6.1 top m andof volute 1:50, chamber respectively. Thereis are 0.72 no m. ventilation The diameter holes and depth in the of volute the vertical chamber, shaft gradientare 0.216 m section, and 1.82 or m, vertical respectively. shaft, The and depth air flows of the dissipation well is 0.28 m and the height of gradient section is 0.24 m. The outlet tunnel has an with the waterWater 2018 from, 10, x the FOR intakePEER REVIEW tunnel into the vertical shaft. The sloped section and the3 straightof 19 plate arched shape, with a width, height, length and slope of 0.2 m, 0.24 m, 6.1 m and 1:50, respectively. are set at the inlet of the outlet tunnel to decrease fluctuations in the water and ensure that the flow Therechanged are no ventilation from 0.2 m ×holes 0.24 min (width the volute × height) chamber, to 0.176 gradient m × 0.33 section,m (width or × height).vertical The shaft, diameter and air of flows regimewith is the open-channelthe watervolute fromchamber the flow. is intake 0.352 The m,tunnel structureand the into height the is shownvertic from theal shaft.in gradient Figure The section 1sloped. To to reveal sectionthe top the ofand volute characteristics the straightchamber plate of the energyare dissipation setis at0.72 the m. inlet The and diameterof thethe cavitationoutlet and depth tunnel of within theto decreasevertical the shaft vortex fluctuations are drop0.216 m shaft,in and the 1.82 threewater m, respectively. testsand ensure at flood The that depth frequencies the flow of 5%, 2%regime andof isthe 0.1% open-channel dissipation (expressed well flow. is as 0.28 PThe =m 5%,andstructure the 2% height and is shown 0.1%of gradient inin theFigure section following) 1. is To 0.24 reveal m. wereThe the outletcarried characteristics tunnel out has in an ofthis the paper. A fluviographenergyarched dissipation (the shape, margins withand athe width, of cavitation error height, are within pluslength or andthe minus slopevortex of 0.18 drop 0.2 mm)m, shaft, 0.24 is m, three used 6.1 mtests to and control at 1:50, flood respectively. the frequencies upstream of water There are no ventilation holes in the volute chamber, gradient section, or vertical shaft, and air flows 5%, 2% and 0.1% (expressed as P = 5%, 2% and 0.1% in the following) were carried out in this paper. level, andwith the tailwaterthe water from depth the intake in the tunnel outlet into tunnel the vertic isal controlled shaft. The sloped by a section valve and located the straight in the plate downstream channel.A fluviograph Eighteenare set at the(the measurement inlet margins of the outletof error points tunnel are plusto are decrease or established minus fluctuations 0.18 inmm) thein isthe used wall water to of andcontrol the ensure vertical the that upstream the shaft, flow water as shown in Figurelevel,1 regime,and the the odd is tailwateropen-channel and even depth flow. numbers in Thethe structureoutlet located tunnel is shown upstream is controlled in Figure and 1. by downstream To a revealvalve thelocated characteristics of thein the vortex downstream of the drop shaft, channel. Eighteen measurement points are established in the wall of the vertical shaft, as shown in respectively;energy the dissipation measurement and the locations cavitation within are close the vortex to wall. drop shaft, three tests at flood frequencies of Figure5%, 1, 2%the and odd 0.1% and (expressed even numbers as P = 5%, located 2% and upstre 0.1% inam the and following) downstream were carried of the out vortex in this paper.drop shaft, In the present study, the experimental velocity was measured by a small specially designed respectively;A fluviograph the measurement (the margins of locations error are plusare close or mi nusto wall. 0.18 mm) is used to control the upstream water L-shapedIn tubelevel, the and [present27 ]the similar tailwater study, to thedepth a Pitot experimental in the tube, outlet which tunnelvelocity is is 160controlled was mm measured longby a valve and by locateda 2 small mm in inspecially the diameter downstream designed with L- a short inletshaped as shownchannel. tube in [27]Eighteen Figure similar 2measurement. to The a Pitot margins pointstube, ofwhichare error established is are160 plusmm in the long or wall minus and of the 2 mm 5vertical percent. in diametershaft, Theas shown with L-shaped ina short tube Figure 1, the odd and even numbers located upstream and downstream of the vortex drop shaft, was alignedinlet as shown withthe in Figure direction 2. The of margins incoming of error flow ar bye plus visual or minus measurement, 5 percent. andThe L-shaped the result tube was was carried respectively; the measurement locations are close to wall. out afteraligned the with liquidIn the the present columndirection study, height of theincoming experimental reached flow the velocityby maximumvisual was measurement, measured and was by a and stable.small the specially result Time-averaged designedwas carried L- pressureout was measuredafter theshaped liquid by tube acolumn liquid[27] similar height column to areached Pitot manometer tube, the which maxi basedismum 160 mm and on long hydrostaticwas and stable. 2 mm Time-averaged principle,in diameter with and pressure a the short accuracy was is 0.1mm.measured Theinlet aeration asby shown a liquid in concentration Figurecolumn 2. Themanometer margins was measured ofbased error onare hy plus withdrostatic or minus a CQ6-2004 principle, 5 percent. aerationand The theL-shaped accuracy concentration tube is was 0.1mm. meter (ChinaThe Institute aerationaligned with ofconcentration Water the direction Resources was of incoming measured and Hydropowerflow with by visual a CQ6-2004 measurement, Research, aeration and Beijing, theconcentration result China), was carried andmeter aout single-chip(China Instituteafter ofthe Waterliquid columnResources height and reached Hydropower the maximum Research, and was stable.Beijing, Time-averaged China), and pressure a single-chip was microcomputer was used to acquire and process the data. The aeration concentration is determined by microcomputermeasured by was a liquid used column to acquire manometer and process based on the hy data.drostatic The principle, aeration and concentration the accuracy is 0.1mm.determined detectingby detecting theThe resistanceaeration the resistance concentration of water of water and was aerated andmeasured aerated water with wa a betweenter CQ6-2004 between two aeration two electrodes. electrodes. concentration The The resolution meterresolution (China is 0.1% 0.1% and Institute of Water Resources and Hydropower Research, Beijing, China), and a single-chip the samplingand the sampling frequency frequency is 1020 is Hz. 1020 Hz. microcomputer was used to acquire and process the data. The aeration concentration is determined by detecting the resistance of water and aerated water between two electrodes. The resolution is 0.1% 2.54 diversion sill x 0.200 and the sampling frequency is 1020 Hz. 2.42

volute2.54 chamber y intakediversion tunnel sill x 0.200 2.04 2.42

1.90 AA i=1:7.5 1 2 volute1.82 chamber y 4 2.04 intake tunnel 0 1 1.70 gradient section 0.176 . 3 4 0 1.90 AA i=1:7.5 R 1 2 1.581.82 4 1.50 0 8R 56 .110 1.70 gradient section 0.176R0 . 0 3 4 0 .1 R 7 1.58 6 1.30 1.50 78 8R 56 R0.10 0 .1 vertical shaft 76 1.101.30 91078 0.200 vertical shaft 1.10 0.200 0.90 11910 12 i=1:50 0.90 0.70 11 12 13 14 B i=1:50 0.70 13 14 B outlet tunnel z 0.30 15 16 outlet tunnel 0.30 15 16dissipation well z 0.14 B 17 18 dissipation well B 0.14 17 18 0.00 y 0.00 y FigureFigureFigure 1. 1. 1.Vortex Vortex Vortex drop shaftshaft shaft profile. profile. proﬁle.

0.0080.008 m m

0.002 m 0.002 m 0.16 m 0.16 m

Figure 2. The structure of L-shaped tube. FigureFigure 2. 2.The The structurestructure of of L-shaped L-shaped tube. tube. Water 2018, 10, 1393 4 of 18

3. Numerical Simulation

3.1. Turbulence Model Considering the advantage of the free interface capture and the specialized air entrainment model inside [28–30], Flow-3D (Flow Science, Santa Fe, NM, USA) was chosen for the numerical simulation. The RNG k-ε turbulence model and the volume of fluid (VOF) method [31] are used for the turbulence model and free surface tracking, respectively. Numerical discretization is performed with the finite difference method. The algebraic equations are solved using the generalized minimum residual (GMRES) method. The fluid is assumed to be incompressible, the governing equations are presented as follows: Continuity equation ∂u i = 0 (1) ∂xi Momentum equation ! 1 ∂p ∂ ∂ui ∂ui ∂ui fi − + ν = + uj (2) ρ ∂xi ∂xj ∂xj ∂t ∂xj k equation ∂(ρk) ∂(ρuik) ∂ ∂k + = αkµe f f + Gk + ρε (3) ∂t ∂xi ∂xi ∂xi ε equation ∗ 2 ∂(ρε) ∂(ρuiε) ∂ ∂ε C1ε ε + = αεµe f f + Gk − C2ερ (4) ∂t ∂xi ∂xi ∂xi k k The fluid configurations are defined in terms of a VOF function F(x,y,z,t), which represents the VOF per unit volume and satisfies the following equation:

∂F ∂ + (Fui) = 0 (5) ∂t ∂xi where ui and xi are the time-averaged velocity and coordinate components, respectively. fi is gravity component, t is the time, µ, ν and ρ are coefficient of dynamic viscosity, kinematic viscosity and density, respectively. Gk is the generation of turbulent energy caused by the average velocity gradient, p is the gauge pressure, µe f f is the revisionary coefficient of dynamic viscosity, F is the VOF function, η ∂u 2 η 1− 1 ∂ui j ∂ui k ∗ η0 2 k µ = µ + µt, G = µt + , µt = ρCµ , C = C − , η = 2E × E , E = e f f k ∂xj ∂xi ∂xj ε 1ε 1ε 1+βη3 ij ij ε ij ∂u 1 ∂ui + j . There are some constants provided by Launder [32] and verified by a large number of 2 ∂xj ∂xi experiments: Cµ = 0.0845, αk = αε = 1.39, C1ε = 1.42, C2ε = 1.68, η0 = 4.377 and β = 0.012.

3.2. Air Entrainment Model The air entrainment model in FLOW-3D was ﬁrst proposed by Hirt in 2003 [33,34]. This model assumes that air entrainment at the free surface is caused by an instability force Pt produced by the turbulence of the free surface. When the level of turbulence exceeds the stability force Pd, which is associated with gravity and surface tension, air with a volume δV will be entrained into the water. The governing equations are as follows:

CNU0.75k1.5 L = (6) T ε σ Pt = ρk; Pd = ρgn LT + (7) LT Water 2018, 10, 1393 5 of 18

h i ( 2(Pt−Pd) C A i f Pt > P δV = air S ρ d (8) 0 i f Pt < Pd Water 2018, 10, x FOR PEER REVIEW 5 of 19 where LT denotes the turbulence length scale, CNU is a constant equal to 0.09, k and ε are the turbulent kinetic energy and turbulent dissipation rate,2𝑃 respectively.−𝑃 gn is the component of gravity normal to 𝐶 𝐴 𝑖𝑓𝑃𝑃 the free surface, σ is the coefﬁcient𝛿𝑉 of = surface tension,𝜌 δV is the volume of air entrained(8) per unit time, 0 𝑖𝑓 𝑃 𝑃 Cair is a coefﬁcient of proportionality and As represents the surface area.

where 𝐿 denotes the turbulence length scale, 𝐶𝑁𝑈 is a constant equal to 0.09, 𝑘 and 𝜀 are the 3.3. Computational Grid and Boundary Conditions turbulent kinetic energy and turbulent dissipation rate, respectively. gn is the component of gravity normal to the free surface, 𝜎 is the coefficient of surface tension, 𝛿𝑉 is the volume of air entrained per For comparison with the experimental results, the upstream (Y-Min) and top (Z-Max) boundaries unit time, 𝐶 is a coefficient of proportionality and 𝐴 represents the surface area. are set as pressure boundaries, and the water elevation at the upstream boundary is added, the downstream3.3. Computational (Y-Max) Grid and boundary Boundary Conditions is set as the outflow. The left (X-Min), right (X-Max) and bottom (Z-Min) boundariesFor comparison are setwith as the solid, experimental non-slip wallresults, boundaries. the upstream Moreover, (Y-Min) and the boundariestop (Z-Max) of the nested grids areboundaries set as symmetry,are set as pressure as shown boundaries, in Figure and 3tha.e Thewater model elevation includes at the upstream 2 mesh boundary blocks, andis the nested added, the downstream (Y-Max) boundary is set as the outflow. The left (X-Min), right (X-Max) and grids are used to ensure the accuracy of grid segmentation in the calculation, as shown in Figure3b. bottom (Z-Min) boundaries are set as solid, non-slip wall boundaries. Moreover, the boundaries of To evaluatethe nested the grids grid are independence set as symmetry, of as the shown results, in Figure the 3a. discharges The model testsincludes were 2 mesh undertaken blocks, and with different grids.the Four nested schemes grids are were used carriedto ensure out the accuracy to verify of gridgrid segmentation accuracy in inTable the calculation,1, the difference as shown in between grids 3 and 4Figure was 3b. 1.12%, To evaluate which the suggested grid independence that grid of the 4 was results, already the discharges sufficient tests for were the undertaken simulation. The time with different grids. Four schemes were carried out to verify grid accuracy in Table 1, the difference step is variable, and the initial time step and residual error are 1 × 10−7 and 1 × 10−6, respectively. between grids 3 and 4 was 1.12%, which suggested that grid 4 was already sufficient for the Additionally,simulation. the The termination time step is variable, time is and set asthe theinitial time time at step which and residual the rate error of changeare 1 × 10 in−7 and the 1 total× volume of the fluid10−6 is, respectively. less than 0.1%,Additionally, and simulation the termination time time is is 200 set sas in the this time paper. at which the rate of change in the total volume of the fluid is less than 0.1%, and simulation time is 200 s in this paper.

(a) Boundary conditions

(b) Grid mesh

Figure 3. Boundary conditions and grid mesh in the numerical model. Water 2018, 10, x FOR PEER REVIEW 6 of 19 Water 2018, 10, 1393 6 of 18 Figure 3. Boundary conditions and grid mesh in the numerical model.

Table 1.1. The results ofof gridgrid independenceindependence (P(P == 0.1%).

Grid GridDescription Description Size Size (X (X × YY ×× Z)Z) Discharge Discharge (L/s) (L/s) containingcontaining block block 15 15 mm mm ×× 1515 mm mm ×× 1515 mm mm 1 1 54.4754.47 nested nestedblock block 10 10 mm mm ×× 1010 mm mm ×× 1010 mm mm containingcontaining block block 10 10 mm mm ×× 1010 mm mm ×× 1010 mm mm 2 2 58.2558.25 nested nestedblock block 5 5mm mm ×× 5 5mm mm ×× 5 5mm mm containingcontaining block block 5 5mm mm ×× 5 5mm mm ×× 5 5mm mm 3 3 60.2860.28 nested nestedblock block 3 3mm mm ×× 3 3mm mm ×× 3 3mm mm containingcontaining block block 4 4mm mm ×× 4 4mm mm ×× 4 4mm mm 4 4 × × 60.9660.96 nested nestedblock block 2 2mm mm ×2 2mm mm × 2 2mm mm

4. Results and Discussion

4.1. Model Verification Verification Liu et et al. al. [19] [19 ]found found that that both both the the standard standard and and RNG RNG k-εk turbulence-ε turbulence models models could could simulate simulate the theflood flood movement movement of the of vortex the vortex drop dropshaft shaftspillway spillway well and well reproduced and reproduced the water the waterflow state flow of state the ofswirling the swirling flow cavity flow in cavity the vertical in the verticalshaft; however, shaft; however, the RNG the k-εRNG turbulencek-ε turbulence model predicted model predicted pressure pressureand flow andvelocity flow better velocity than better the standard than the k- standardε model. kFigure-ε model. 4 shows Figure that4 showsthe pressure that the and pressure velocity anddistributions velocity distributionsof the vortex ofdrop the shaft vortex calculated drop shaft using calculated the two using models the have two similar models patterns. have similar The patterns.pressure Theand pressurevelocity values and velocity calculated values in calculatedthe vertical in shaft the vertical were close shaft and were inclose rather and good in rather agreement good agreementwith the experiment; with the experiment; however, however,the RNG thek-ε RNGturbulencek-ε turbulence model generated model generated results resultscloser closerto those to thoseobtained obtained in the in experiment, the experiment, especially especially in the in dissip the dissipationation well. well. The Therelative relative deviations deviations for forpressure pressure of ofthe the standard standard and and RNG RNG k-εk -turbulenceε turbulence models models with with respect respect to to the the model model test test results results were were 18% and 11%, respectively, andand thethe relative deviations for velocity were 8% and 5%, respectively. In general, the RNG k-ε turbulenceturbulence modelmodel cancan better simulate the flowflow of the vortex drop shaft than the standard k-ε turbulenceturbulence model,model, indicatingindicating that that the the RNG RNGk k-ε-εturbulence turbulence model model is is more more suitable suitable for for dealing dealing with with a distorteda distorted flow flow or or rotational rotational flow flow with with a higha high strain stra rate.in rate. Therefore, Therefore, the the following following analysis analysis isbased is based on theon the RNG RNGk-ε kmodel.-ε model.

2.0 2.0 z (m) experiment z (m) experiment

RNG RNG 1.5 1.5 standard standard 1.0 1.0

0.5 0.5 pressure pressure (kPa) (kPa) 0.0 0.0 -2.0 2.0 6.0 10.0 -2.0 2.0 6.0 10.0 (a) Upstream pressure (b) Downstream pressure

Figure 4. Cont. Water 2018, 10, 1393 7 of 18

WaterWater 2018 2018, ,10 10, ,x x FOR FOR PEER PEER REVIEW REVIEW 77 of of 19 19

2.02.0 z (m) 2.02.0 z (m) experimentexperiment zz (m) (m) experimentexperiment RNGRNG RNGRNG 1.51.5 1.51.5 standardstandard standardstandard 1.01.0 1.01.0

0.50.5 0.50.5 velocityvelocity velocityvelocity (m/s)(m/s) (m/s)(m/s) 0.00.0 0.00.0 0.0 2.0 4.0 6.0 0.0 2.0 4.0 6.0 0.0 2.0 4.0 6.0 0.0 2.0 4.0 6.0 ((cc)) Upstream Upstream velocity velocity ( (dd)) Downstream Downstream velocity velocity

FigureFigure 4. 4. Pressure Pressure and and velocity velocity calculated calculated using using different different turbulence turbulenceturbulence models modelsmodels (P (P(P = == 0.1%). 0.1%). 4.2. Flow Regime 4.2.4.2. Flow Flow Regime Regime The discharges, which depend on the flood elevation upstream, are obtained through via flow TheThe discharges, discharges, which which depend depend on on the the flood flood elev elevationation upstream, upstream, are are obtained obtained through through via via flow flow meter and baffle (the baffle is a tool in Flow-3D). Because of measurement error in flow meter and metermeter andand bafflebaffle (the (the bafflebaffle isis a a tooltool inin Flow-3 Flow-3D).D). BecauseBecause ofof measurementmeasurement errorerror inin flow flow metermeter andand computational grid, small differences were unavoidable. Overall, the errors between experiment and computationalcomputational grid, grid, small small differences differences were were unavoida unavoidable.ble. Overall, Overall, the the errors errors between between experiment experiment and and simulation were less than 3% (the maximum error is 2.14% at the flood frequency of 0.1%) as shown simulationsimulation were were less less than than 3% 3% (the (the maximum maximum error error is is 2.14% 2.14% at at the the flood flood frequency frequency of of 0.1%) 0.1%) as as shown shown in Table2. The numerical results were similar to the experimental results, which suggested that the inin Table Table 2. 2. The The numerical numerical results results were were similar similar to to the the experimental experimental results, results, which which suggested suggested that that the the numerical simulation results were reliable. Figure5 shows the Froude number (the intake tunnel has an numericalnumerical simulation simulation results results were were reliable. reliable. Figure Figure 5 5 shows shows the the Froude Froude number number (the (the intake intake tunnel tunnel has has archedan arched shape shape and and the flowthe flow was non-pressurewas non-pressure flow; fl becauseow; because the flow the can flow be can viewed be viewed as the open-channel as the open- an arched shape and the flow was non-pressure flow; because the flow can bep viewed as the open- flow in a rectangular channel, the Froude number is calculated by Fr = V/𝐹 g= h, where h is water channelchannel flowflow inin aa rectangularrectangular channel,channel, thethe FroudeFroude numbernumber isis calculatedcalculated byby 𝐹 = V/V/gg h h, , wherewhere hh isis depth) in the intake tunnel. The Froude number was larger than 1 and increased with the discharge, waterwater depth)depth) inin thethe intakeintake tunnel.tunnel. TheThe FroudeFroude nunumbermber waswas largerlarger thanthan 11 andand increasedincreased withwith thethe which indicated the flow was draining freely into the drop shaft and was supercritical along the entire discharge,discharge, whichwhich indicatedindicated thethe flowflow waswas drainingdraining freelyfreely intointo thethe dropdrop shaftshaft andand waswas supercriticalsupercritical intake tunnel. Even though there were some errors between experiment and simulation due to the alongalong thethe entireentire intakeintake tunnel.tunnel. EvenEven thoughthough ththereere werewere somesome errorserrors betweenbetween experimentexperiment andand surface fluctuations after aeration and unavoidable instrument error, in general, the numerical model simulationsimulation duedue toto thethe surfacesurface fluctuationsfluctuations afterafter aerationaeration andand unavoidableunavoidable instrumentinstrument error,error, inin was able to model the main characteristics of the vortex drop shaft. general,general, the the numerical numerical model model was was able able to to model model the the main main characteristics characteristics of of the the vortex vortex drop drop shaft. shaft.

Table 2. Comparison of discharge between experiment and simulation. TableTable 2. 2. Comparison Comparison of of discharge discharge between between experiment experiment and and simulation. simulation. Discharge (L/s) Flood Frequency DischargeDischarge (L/s) (L/s) Deviation FloodFlood Frequency Frequency DeviationDeviation ExperimentExperimentExperiment SimulationSimulationSimulation 5%5% 5% 30.4530.4530.45 29.9229.9229.92 1.74% 1.74%1.74% 2%2% 2% 38.2138.2138.21 38.0138.0138.01 0.52% 0.52%0.52% 0.1% 62.29 60.96 2.14% 0.1%0.1% 62.2962.29 60.9660.96 2.14%2.14%

Figure 5. Froude number profile in intake tunnel. FigureFigure 5. 5. Froude Froude number number profile profile in in intake intake tunnel. tunnel. Water 2018, 10, 1393 8 of 18 WaterWater 2018 2018, ,10 10, ,x x FOR FOR PEER PEER REVIEW REVIEW 88 of of 19 19 In the experiment, the water entered the intake tunnel in a state of open-channel flow, and it InIn thethe experiment,experiment, thethe waterwater enteredentered thethe intakeintake tutunnelnnel inin aa statestate ofof open-channelopen-channel flow,flow, andand itit provided a height of 5.0-cm for the air inlet under the flood frequency of 0.1% (as shown in Figure6). providedprovided aa heightheight ofof 5.0-cm5.0-cm forfor thethe airair inletinlet underunder ththee floodflood frequencyfrequency ofof 0.1%0.1% (as(as shownshown inin FigureFigure 6).6). The discharged water flowed into the vertical shaft with a large amount of air to form a stable mixing TheThe dischargeddischarged waterwater flowedflowed intointo thethe verticalvertical shaftshaft withwith aa largelarge amountamount ofof airair toto formform aa stablestable mixingmixing cavity through the intake tunnel. The water surface at the downstream wall of the vortex chamber cavitycavity throughthrough thethe intakeintake tunnel.tunnel. TheThe waterwater surfacsurfacee atat thethe downstreamdownstream wallwall ofof thethe vortexvortex chamberchamber was higher than that at the upstream wall, and the surface exhibited some fluctuations, as shown in waswas higherhigher thanthan thatthat atat thethe upstreamupstream wall,wall, andand ththee surfacesurface exhibitedexhibited somesome flfluctuations,uctuations, asas shownshown inin Figures6 and7. The cavity throat (i.e., the narrowest cavity section) occurred in the gradient section FiguresFigures 66 andand 7.7. TheThe cavitycavity throatthroat (i.e.,(i.e., thethe narrowenarrowestst cavitycavity section)section) occurredoccurred inin thethe gradientgradient sectionsection because of the decreases in the section size and discharge area. As the axial velocity increased in the becausebecause ofof thethe decreasesdecreases inin thethe sectionsection sizesize andand dischargedischarge area.area. AsAs thethe axialaxial velocityvelocity increasedincreased inin thethe vertical shaft, the water layer became thinner. Then, water dropped into the dissipation well at the end verticalvertical shaft,shaft, thethe waterwater layerlayer becamebecame thinner.thinner. Then,Then, waterwater droppeddropped intointo thethe dissipationdissipation wellwell atat thethe of the vertical shaft and formed a water cushion, which exhibited a large-scale swirling motion and endend ofof thethe verticalvertical shaftshaft andand formedformed aa waterwater cushiocushion,n, whichwhich exhibitedexhibited aa large-scalelarge-scale swirlingswirling motionmotion intense mixing. An aeration phenomenon was obvious, and the flow was full of milky white bubbles. andand intenseintense mixing.mixing. AnAn aerationaeration phenomenonphenomenon waswas obvious,obvious, andand thethe flowflow waswas fullfull ofof milkymilky whitewhite Highly turbulent flow was observed in the sloped section, air bubbles gradually drifted upward and bubbles.bubbles. HighlyHighly turbulentturbulent flowflow waswas observedobserved inin thethe slopedsloped section,section, airair bububblesbbles graduallygradually drifteddrifted out from the surface in the outlet tunnel, and the flow was stable. upwardupward andand outout fromfrom thethe surfacesurface inin thethe outletoutlet tunnel,tunnel, andand thethe flowflow waswas stable.stable.

((aa)) CavityCavity inin thethe dropdrop shaftshaft ((bb)) DissipationDissipation wellwell

FigureFigure 6.6. FlowFlowFlow conditionsconditions inin thethe experimentexperiment (P(P == 0.1%).0.1%). 0.1%).

((aa)) LongitudinalLongitudinal profileprofile ( (bb)) TransverseTransverse profileprofile

FigureFigureFigure 7.7. 7. VolumeVolumeVolume fractionfraction ofof fluidfluid ﬂuid inin thethe numericalnumerical simulation simulation (P (P = = 0.1%).0.1%).

4.3.4.3. VelocityVelocity Velocity DistributionDistribution TheThe flowflow flow velocityvelocity isis anan importantimportant parameterparameter whwh whenenen analyzinganalyzing thethe the energyenergy dissipationdissipation effecteffect effect andand calculatingcalculating thethe the energyenergy energy dissipationdissipation ratio.ratio. TheThe The velocityvelocity velocity inin in thethe the vortexvortex vortex dropdrop drop shaftshaft shaft waswas was mainlymainly mainly affectedaffected affected byby by gravity,gravity, frictionfriction andand shearshear force,force, whichwhich firstfirst first incrincr increasedeasedeased andand thenthen decreased,decreased, andand thethe velocityvelocity peakedpeaked atat thethe connectionconnection withwith thethe slopedsloped sectionsection (z(z == 0.700.70 m).m). TheThe maximummaximum velocitiesvelocities werewere 2.502.50 m/s,m/s, 2.682.68 m/s,m/s, 3.453.45 m/sm/s (at(at floodflood frequenciesfrequencies ofof 5%,5%, 2%2% andand 0.1%,0.1%, respectively)respectively) inin thethe experimentexperiment andand 2.562.56 m/s,m/s, Water 2018, 10, 1393 9 of 18 at the connection with the sloped section (z = 0.70 m). The maximum velocities were 2.50 m/s, Water 2018, 10, x FOR PEER REVIEW 9 of 19 2.68 m/s, 3.45 m/s (at flood frequencies of 5%, 2% and 0.1%, respectively) in the experiment and 2.56 m/s, 2.89 m/s, 3.49 m/s in the simulation, as shown in Figure8. At the bottom of the vertical 2.89 m/s, 3.49 m/s in the simulation, as shown in Figure 8. At the bottom of the vertical shaft, the shaft, the upper and lower flows collided, resulting in a rapid decrease in the velocity, indicating that upper and lower flows collided, resulting in a rapid decrease in the velocity, indicating that the the energy dissipation effect was sufficient. energy dissipation effect was sufficient.

2.0 z (m) 0.1% experiment 0.1% simulation 2% experiment 1.5 2% simulation 5% experiment 5% simulation 1.0

0.5 velocity (m/s) 0.0 0.0 1.0 2.0 3.0 4.0 5.0 (a) Upstream velocity 2.0 z (m) 0.1% experiment 0.1% simulation 2% experiment 1.5 2% simulation 5% experiment 5% simulation 1.0

0.5

velocity (m/s) 0.0 0.0 1.0 2.0 3.0 4.0 5.0 (b) Downstream velocity

FigureFigure 8. 8. VelocityVelocity in in the the vortex vortex drop drop shaft. shaft.

TheThe velocity fieldfield (flood(flood frequency frequency of of 0.1%) 0.1%) is shown is shown in Figure in Figure9. At the9. At same the elevation, same elevation, the velocity the velocitycloser to closer the wall to the was wall smaller was duesmaller to friction due to alongfriction the along vertical the shaftvertical wall. shaft When wall. the When water the fell water from fellthe from wall intothe thewall dissipation into the dissipation well, the maximum well, the velocity maximum was velocity approximately was approximately 5.0 m/s. Subsequently, 5.0 m/s. Subsequently,the discharged the flow discharged collided with flow the collided water cushion with the in thewater well, cushion and the in velocity the well, rapidly and the decreased velocity to rapidly1.0 m/s decreased at the bottom to 1.0 of m/s the wellat the and bottom approximately of the well 2.0 and m/s approximately at the tunnel 2.0 outlet. m/s Thus,at the thetunnel energy outlet. was Thus,effectively the energy dissipated. was effectively dissipated. The direction of the velocity varied within the vertical shaft, and the average velocity was calculated based on a theoretical analysis to quantify the cross-sectional kinetic energy distribution. The velocity of a swirling flow can be decomposed into a tangential velocity, an axial velocity and a radial velocity. The radial velocity is often ignored because it is smaller than the tangential and axial velocities (generally by more than two orders of magnitude). Additionally, it is often assumed that swirling flow is associated with a free eddy and meets the conditions for the conservation of angular momentum. Water 2018, 10, 1393 10 of 18 Water 2018, 10, x FOR PEER REVIEW 10 of 19

(a) Longitudinal profile (b) Transverse profile

FigureFigure 9. 9. ResultantResultant velocity velocity field ﬁeld in in the the numerical numerical simulation simulation (P (P = = 0.1%). 0.1%).

TheTwo direction sections of in the velocity vertical shaftvaried are within used th toe establishvertical shaft, the Bernoulli and the equation,average velocity as shown was in calculatedFigure 10: based on a theoretical analysis to quantify the cross-sectional kinetic energy distribution. The velocity of a swirling flow canV be2 decomposed(V + dVinto)2 a tangentialdS V2 velocity, an axial velocity and a + dZ = + λ (9) radial velocity. The radial velocity2 isg often ignored2 gbecause it 4isR smaller0 2g than the tangential and axial velocities (generally by more than two orders of magnitude). Additionally, it is often assumed that where the hydraulic radius is R0 = A = Q and the water trajectory is dS = dZ ; R and Q swirling flow is associated with a free2 πeddyR and2πRV meetcos θ s the conditions for the conservationcos ofθ angular are the radius of vertical shaft and the flow discharge, respectively; g is the constant of gravitational momentum. acceleration; θ is the angle between the direction of the velocity and the vertical direction; and λ Two sections in the vertical shaft are used to establish the Bernoulli equation, as shown in Figure denotes the factor of friction loss. Additionally, V is the resultant velocity, and dV is considered 10: negligible in this study. The hydraulic radius and the water trajectory are substituted into Equation (9) to obtain Equation (10): 𝑉 𝑉+𝑑𝑉 𝑑𝑆 𝑉 +𝑑Z= +𝜆 (9) 2g V2 2g πλR V4𝑅3 2g d = 1 − dZ (10) 2g 2Q 2g where the hydraulic radius is 𝑅 = = and the water trajectory is 𝑑𝑆 = ; 𝑅 and 𝑄 are 3/2 2/3 the radius3/2 of πλverticalR V2 shaft and the flowπλR discharge, respectively;V2 g is the constant of gravitational with G = 2Q 2g and K = 2Q , we have G = K 2g . This formula can be substituted into acceleration;Equation (10), 𝜃 is and the the angle new between equation the can direction then be integrated of the velocity as follows: and the vertical direction; and 𝜆 denotes the factor of friction loss. Additionally, 𝑉 is the resultant velocity, and 𝑑𝑉 is considered Z Z Z G dG Z G dG Z G0 dG negligible in this study.K Thed hydraulic= radius and= the water trajectory− are substituted into Equation Z 3/2 3/2 3/2 (11) (9) to obtain Equation (10):0 G0 1 − G 0 1 − G 0 1 − G 𝑉 H(G) = H(G𝜋𝜆𝑅0) +𝑉KZ (12) 𝑑 =1− 𝑑Z (10) √ ! 2 1 1 2g √ 2𝑄22g 2 G + 1 1 H(G) = ln √ + ln G + G + 1 − √ arctan √ − arctan √ (13) / 3 1 − G 3 / 3 3 3 with 𝐺/ = and 𝐾= , we have 𝐺=𝐾 . This formula can be substituted into g g where K is a coefficient, G is a function related to R, Q, λ and V, H is a function related to G, Z is the Equation (10), and the new equation can then be integrated as follows: length of the shaft from the gradient section. The initial velocity was obtained by𝑑𝐺 measuring the velocity𝑑𝐺 in the vertical 𝑑𝐺 shaft (at z = 0, as shown 𝐾 𝑑Z= = − (11) in Figure 10) several times. Then, the1−𝐺 average/ velocity1−𝐺 in each/ section was1−𝐺 calculated/ by combining Equations (12) and (13). In the numerical simulation, the average velocity was measured by establishing a flux plane (baffle) in the measurement section. The average velocities obtained based on the theoretical calculations and simulations were compared in Figure 11. This figure shows that the 𝐻𝐺 =𝐻𝐺 +𝐾Z (12) analytically and numerically obtained resultant velocities were in good agreement. Both results also had similar patterns of variation and the mean error between the theoretical and simulated values were 3.81%, 3.63% and 5.65%2 (at flood1 frequencies1 of 5%, 2%2 and 0.1%, respectively).2√𝐺 +1 In general,1 there was a 𝐻𝐺 = 𝑙𝑛 + 𝑙𝑛𝐺 + √𝐺 +1− 𝑎𝑟𝑐𝑡𝑎𝑛 −𝑎𝑟𝑐𝑡𝑎𝑛 (13) 3 1−√𝐺 3 √3 √3 √3 Water 2018, 10, x FOR PEER REVIEW 11 of 19 where 𝐾 is a coefficient, 𝐺 is a function related to 𝑅, 𝑄, 𝜆 and 𝑉, 𝐻 is a function related to 𝐺, Z is the length of the shaft from the gradient section.

2r 0 Vt

dz V θ s Vz d Water 2018, 10, 1393 11 of 18 V+dV Water 2018, 10, x FOR PEER REVIEW z 11 of 19 highwhere consistency 𝐾 is a coefficient, between 𝐺 the is a theoretical function related and numerical2R to 𝑅, results.𝑄, 𝜆 and 𝑉, The 𝐻 is developed a function theoretical related to calculation 𝐺, Z is the methodlength of for the ﬂow shaft velocity from the can gradient function section. as a reference for vertical shaft design.

2r 0 Vt

Figure 10. Diagram of thedz average velocity in the shaft section. V θ s Vz d The initial velocity was obtained by measuringV+dV the velocity in the vertical shaft (at z = 0, as shown in Figure 10) several times. Then, the average velocity in eachz section was calculated by combining Equations (12) and (13). In the numerical simulation, the average velocity was measured by establishing a flux plane (baffle) in the measurem2R ent section. The average velocities obtained based on the theoretical calculations and simulations were compared in Figure 11. This figure shows that the analytically and numerically obtained resultant velocities were in good agreement. Both results also had similar patterns of variation and the mean error between the theoretical and simulated values were 3.81%, 3.63% and 5.65% (at flood frequencies of 5%, 2% and 0.1%, respectively). In general, there was a high consistency between the theoretical and numerical results. The developed theoretical calculationFigure method 10. Diagramfor flow ofvelocity the average can function velocityvelocity in as the a reference shaft section. for vertical shaft design.

The initial2.0 velocityz (m) was obtained by measuring the velocity in the0.1% vertical theoretical shaft (at z = 0, as shown in Figure 10) several times. Then, the average velocity in each section0.1% was simulation calculated by combining 2% theoretical Equations (12)1.5 and (13). In the numerical simulation, the average2% velocity simulation was measured by establishing a flux plane (baffle) in the measurement section. The average5% theoreticalvelocities obtained based 5% simulation on the theoretical1.0 calculations and simulations were compared in Figure 11. This figure shows that the analytically and numerically obtained resultant velocities were in good agreement. Both results also had similar patterns of variation and the mean error between the theoretical and simulated 0.5 values were 3.81%, 3.63% and 5.65% (at flood frequencies of 5%, 2% and 0.1%,velocity respectively). In general, there was a high consistency between the theoretical and numerical results.(m/s) The developed theoretical calculation0.0 method for flow velocity can function as a reference for vertical shaft design. 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 2.0 0.1% theoretical FigureFigure 11.11.Comparison zComparison (m) of ofthe thesimulated simulatedand andtheoretical theoretical average average velocity velocity in in the the shaft shaft section. section. 0.1% simulation 4.4. Pressure Distribution 2% theoretical 4.4. Pressure1.5 Distribution 2% simulation Cavitation is a phenomenon that may occur when the local pressure5% theoretical extends below the vapor Cavitation is a phenomenon that may occur when the local pressure5% simulation extends below the vapor pressurepressure ofof the1.0 the liquid liquid at at the the operating operating temperature. temperature. WhenWhen cavitationcavitation occurs,occurs, collapsingcollapsing cavitationcavitation bubblesbubbles cancan causecause materialmaterial erosion,erosion, therebythereby decreasingdecreasing thethe lifespanlifespan andand performanceperformance ofof thethe vortexvortex dropdrop shaftshaft spillwayspillway0.5 [[35].35]. Therefore,Therefore, the pressure is one of the the most most important important indicators indicators of of whether whether a avortex vortex drop drop shaft shaft can can stably stably operate; operate; consequently, consequently, negative negative pressure pressure zones zones should shouldvelocity be be avoided avoided or or as as small as possible. The time-averaged pressure along the vortex drop shaft exhibited(m/s) an obvious small as possible.0.0 The time-averaged pressure along the vortex drop shaft exhibited an obvious distribution characteristic, being large at the top and bottom but small at the middle as shown in distribution characteristic,1.0 1.5 being 2.0 large 2.5 at the 3.0 top 3.5 and bottom 4.0 but 4.5 small 5.0 at the 5.5 middle 6.0 as shown in FigureFigure 12 12.. The minimum minimum pressures pressures were were 0.04 0.04 kPa, kPa, 0. 0.1313 kPa, kPa, 0.23 0.23 kPa kPa (at (at flood ﬂood frequencies frequencies of 5%, of 5%, 2% 2% and 0.1%,Figure respectively) 11. Comparison in of the the experiment simulated and and theoretica 0.07 kPa,l average 0.12 kPa, velocity 0.18 in kPa the shaft in the section. simulation at z = 0.7 m. Additionally, the pressures in the gradient section and dissipation well were obviously 4.4. Pressure Distribution higher than that in the vertical shaft. The wall pressure in the vertical shaft was caused by a centrifugal force,Cavitation and during is thea phenomenon falling process, that the may tangential occur wh velocityen the and local centrifugal pressure forceextends gradually below decreased,the vapor whichpressure subsequently of the liquid led at to the the op decreasederating temperature. pressure. When cavitation occurs, collapsing cavitation bubbles can cause material erosion, thereby decreasing the lifespan and performance of the vortex drop shaft spillway [35]. Therefore, the pressure is one of the most important indicators of whether a vortex drop shaft can stably operate; consequently, negative pressure zones should be avoided or as small as possible. The time-averaged pressure along the vortex drop shaft exhibited an obvious distribution characteristic, being large at the top and bottom but small at the middle as shown in Figure 12. The minimum pressures were 0.04 kPa, 0.13 kPa, 0.23 kPa (at flood frequencies of 5%, 2% Water 2018, 10, x FOR PEER REVIEW 12 of 19

and 0.1%, respectively) in the experiment and 0.07 kPa, 0.12 kPa, 0.18 kPa in the simulation at z = 0.7 m. Additionally, the pressures in the gradient section and dissipation well were obviously higher than that in the vertical shaft. The wall pressure in the vertical shaft was caused by a centrifugal force, Water 2018and ,during10, 1393 the falling process, the tangential velocity and centrifugal force gradually decreased,12 of 18 which subsequently led to the decreased pressure.

(a) Upstream pressure

(b) Downstream pressure

FigureFigure 12. 12.Pressure Pressure trends in in the the vertical vertical shaft. shaft.

An areaAn area of negativeof negative pressure pressure formed formed atat thethe connection between between the the vertical vertical shaft shaft and andoutlet outlet tunnel because the swirling flow moved away from the vertical shaft wall as shown in Figure 13 tunnel because the swirling flow moved away from the vertical shaft wall as shown in Figure 13 (flood (flood frequency of 0.1%). Moreover, the pressure in the dissipation well was larger, approximately frequency of 0.1%). Moreover, the pressure in the dissipation well was larger, approximately 10 kPa, 10 kPa, due to the influences of the hydrostatic pressure and discharge. The maximum pressure in due tothe the entire influences vortex drop of the shaft hydrostatic was observed pressure at the and bottom discharge. of the dissipation The maximum well, with pressure experimental in the entire vortexand drop simulated shaft was values observed of 13.1 kPa at the and bottom 12.6 kPa, of respectively the dissipation (flood well, frequency with experimental of 0.1%). The andsimilarity simulated values of 13.1 kPa and 12.6 kPa, respectively (flood frequency of 0.1%). The similarity between the Waterbetween 2018 , 10the, x FORexperimental PEER REVIEW and simulated values further verified the accuracy of the numerical13 of 19 experimentalsimulation. and simulated values further verified the accuracy of the numerical simulation.

(a) Longitudinal profile (b) Transverse profile

FigureFigure 13. 13.Pressure Pressure proﬁles profiles based based on the numerical numerical simulations simulations (P = (P 0.1%). = 0.1%).

4.5. Aeration Concentration In the vortex drop shaft without ventilation holes, the water flowed into the vertical shaft with a large amount of air and created a stable mixing cavity through the intake tunnel. Due to the rotation of the flow and the existence of the cavity, air was continuously drawn into the cavity, and the velocity was high in the cavity area. In the experiment, the water layer was thin, and there was an obvious aeration phenomenon. The aerated water rushed into the dissipation well at a high speed and there was considerable velocity fluctuation in the water cushion. The aeration concentration was measured with a CQ6-2004 aeration concentration meter, and the results are shown in Figure 14.

2.0 z (m) 0.1% experiment 0.1% simulation 2% experiment 1.5 2% simulation 5% experiment 1.0 5% simulation

0.5

C (%) 0.0 0 5 10 15 20 25 30 35 40 (a) Upstream aeration concentration 2.0 z (m) 0.1% experiment 0.1% simulation 1.5 2% experiment 2% simulation 5% experiment 1.0 5% simulation

0.5

C (%) 0.0 0 5 10 15 20 25 30 35 40 (b) Downstream aeration concentration

Figure 14. Aeration concentration in the vortex drop shaft. Water 2018, 10, x FOR PEER REVIEW 13 of 19

Water 2018, 10, 1393(a) Longitudinal profile (b) Transverse profile 13 of 18 Figure 13. Pressure profiles based on the numerical simulations (P = 0.1%).

4.5. Aeration Concentration 4.5. Aeration Concentration In the vortex drop shaft without ventilation holes, the water flowed into the vertical shaft with a In the vortex drop shaft without ventilation holes, the water flowed into the vertical shaft with large amount of air and created a stable mixing cavity through the intake tunnel. Due to the rotation of a large amount of air and created a stable mixing cavity through the intake tunnel. Due to the rotation the flow and the existence of the cavity, air was continuously drawn into the cavity, and the velocity of the flow and the existence of the cavity, air was continuously drawn into the cavity, and the was high in the cavity area. In the experiment, the water layer was thin, and there was an obvious velocity was high in the cavity area. In the experiment, the water layer was thin, and there was an aeration phenomenon. The aerated water rushed into the dissipation well at a high speed and there obvious aeration phenomenon. The aerated water rushed into the dissipation well at a high speed was considerable velocity fluctuation in the water cushion. The aeration concentration was measured and there was considerable velocity fluctuation in the water cushion. The aeration concentration was with a CQ6-2004 aeration concentration meter, and the results are shown in Figure 14. measured with a CQ6-2004 aeration concentration meter, and the results are shown in Figure 14.

2.0 z (m) 0.1% experiment 0.1% simulation 2% experiment 1.5 2% simulation 5% experiment 1.0 5% simulation

0.5

C (%) 0.0 0 5 10 15 20 25 30 35 40 (a) Upstream aeration concentration 2.0 z (m) 0.1% experiment 0.1% simulation 1.5 2% experiment 2% simulation 5% experiment 1.0 5% simulation

0.5

C (%) 0.0 0 5 10 15 20 25 30 35 40 (b) Downstream aeration concentration

FigureFigure 14.14. AerationAeration concentrationconcentration inin thethe vortexvortex dropdrop shaft.shaft.

Aerating a large amount of air at a low-pressure is an effective way to prevent cavitation erosion. When the air content in the water increases, the compressibility of the mixture of water and vapor correspondingly increases, which can mitigate the impact of cavitation collapse and reduce the risk of erosion. Experimental data [36,37] showed that when the aeration concentration in the flow reached 1% to 2%, cavitation erosion at the solid boundary was reduced. Moreover, when the aeration concentration reached 5% to 7%, cavitation erosion would not occur. In this paper, the data showed that the aeration concentration ranged approximately from 5% to 25% and decreased with the increasing discharge, which reflected a low probability of cavitation erosion. However, the values Water obtained2018, 10, 1393 in the experiment were almost larger than the results of the numerical simulation,14 of 18 suggesting that the numerical simulation of aeration concentration needs further study. Because of the effects of surface tension and viscosity, the aeration concentration is underestimated so that the 4.6. Cavitationaeration concentration Number is larger in the prototype [23–26]. Therefore, the results of aeration concentrationThe probability could of be cavitation used as a reference is lower for when engineering there is design a greater difference between the absolute pressure and the vapor pressure of a liquid at a certain point. In addition, the larger the velocity is, 4.6. Cavitation Number the more easily cavitation occurs. Therefore, the cavitation number is often used to describe the degree of ﬂow cavitationThe probability [38] and of cavitation can be deﬁned is lower as when follows: there is a greater difference between the absolute pressure and the vapor pressure of a liquid at a certain point. In addition, the larger the velocity is, the more easily cavitation occurs. Therefore, thep cavitation− p number is often used to describe the N = 0 v (14) degree of flow cavitation [38] and can be defined as0.5 follows:ρV2

𝑝 −𝑝 𝑁= where p0 is the absolute pressure, pv is the elevation-dependent0.5𝜌𝑉 vapor pressure, V is the(14) resultant velocity close to the wall, and ρ is the density. The smaller the cavitation number From Equation (14) where 𝑝 is the absolute pressure, 𝑝 is the elevation-dependent vapor pressure, 𝑉 is the resultant is, the more easily cavitation occurs. velocity close to the wall, and 𝜌 is the density. The smaller the cavitation number From Equation (14) is,In the hydraulic more easily structures, cavitation negative occurs. pressure exists in sections of abrupt changes, thereby increasing the probabilityIn hydraulic of cavitation. structures, In thisnegative study, pressure the body exis changedts in sections abruptly of at abrupt the gradient changes, section thereby and the connectionincreasing between the probability the vertical of cavitation. shaft and outletIn this tunnel,study, the and body therefore changed cavitation abruptly might at the occurgradient in these sections.section The and cavitation the connection number between was calculated the vertical based shaft on and the outlet pressure tunnel, and and velocity, therefore and cavitation the results are shownmight in Figure occur in15 .these sections. The cavitation number was calculated based on the pressure and velocity,Generally, and cavitation the results occursare shown easily in Figure in practical 15. applications when the cavitation number is less than 0.2 [Generally,39]. In this cavitation study, the occurs pressures easily inin thepractical vortex applications chamber, when gradient the cavitation section and number dissipation is less well than 0.2 [39]. In this study, the pressures in the vortex chamber, gradient section and dissipation well were large, and the velocities were small, thus, the associated cavitation numbers were larger than 1.5. were large, and the velocities were small, thus, the associated cavitation numbers were larger than Although negative pressure and large velocity appeared in the vertical shaft and resulted in smaller 1.5. Although negative pressure and large velocity appeared in the vertical shaft and resulted in cavitationsmaller numbers cavitation from numbers 0.40 tofrom 0.80, 0.40 they to 0.80, were they all largerwere all than larger the than critical the valuecritical of value 0.2, soof there0.2, so was a smallerthere probability was a smaller of cavitation.probability of cavitation.

2.0 z (m) 0.1% experiment 0.1% simulation 2% experiment 1.5 2% simulation 5% experiment 5% simulation 1.0

0.5

N 0.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 Water 2018, 10, x FOR PEER REVIEW 15 of 19 (a) Upstream cavitation number

2.0 z (m) 0.1% experiment 0.1% simulation 2% experiment 1.5 2% simulation 5% experiment 5% simulation 1.0

0.5

N 0.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0

(b) Downstream cavitation number

FigureFigure 15. 15.Cavitation Cavitation numbernumber in in the the vortex vortex drop drop shaft. shaft.

4.7. Energy Dissipation Rate The energy dissipation within a vortex drop shaft is mainly reflected by two factors. First, the centrifugal force generated by the swirling flow increases the pressure on the wall and consequently increases the frictional resistance. Simultaneously, as the flow swirls along the wall, the trajectory becomes longer and further increases the head loss. Second, the flow fluctuates with aeration and combined with the large velocity gradient, results in swirling, which causes shearing between flow layers. In addition, the shearing, rotation and collision caused by a flow jet that enters the dissipation well can also dissipate energy. In the present study, the calculated section is shown in Figure 1, where the section of volute chamber A-A at z = 1.9 m and the section of outlet tunnel B-B at Y + 1.8 m. The reference position of the potential energy is z = 0.0 m. The energy dissipation rate of the traditional energy dissipators is generally between 40% and 50%; any rate over 50% is considered excellent. In this paper, the energy dissipation rates exceeded 70% as shown in Table 3, indicating that the vortex drop shaft without ventilation holes sufficiently dissipated energy and met the needs of the actual project. As shown in Figure 16, energy dissipation mainly occurred in the dissipation well and sloped section, and the dissipation rate of turbulent kinetic energy was generally between 20 and 30 m2·s−3 with a maximum of 32 m2·s−3. In addition, the wall friction dissipated some energy, and the dissipation rate of turbulent kinetic energy near the wall in the same cross-section was larger than that in other parts of the structure.

Table 3. The calculation results of the energy dissipation rate.

Section A-A Section B-B Energy Condition Dissipation 𝑽𝟐/𝟐g (m) 𝑯+𝒑/𝜸 (m) 𝑽𝟐/𝟐g (m) 𝑯+𝒑/𝜸 (m) Rate (%) Experiment 0.16 1.65 0.19 0.29 73.48 5% Simulation 0.17 1.69 0.20 0.28 74.12 Experiment 0.18 1.82 0.23 0.31 73.00 2% Simulation 0.19 1.81 0.24 0.33 71.66 Experiment 0.21 2.12 0.24 0.39 72.95 0.1% Simulation 0.25 2.13 0.30 0.41 70.13 Notes: 𝐻 is the elevation head, 𝛾 is the specific weight of water, 𝑝 is the gauge pressure. Water 2018, 10, 1393 15 of 18

4.7. Energy Dissipation Rate The energy dissipation within a vortex drop shaft is mainly reflected by two factors. First, the centrifugal force generated by the swirling flow increases the pressure on the wall and consequently increases the frictional resistance. Simultaneously, as the flow swirls along the wall, the trajectory becomes longer and further increases the head loss. Second, the flow fluctuates with aeration and combined with the large velocity gradient, results in swirling, which causes shearing between flow layers. In addition, the shearing, rotation and collision caused by a flow jet that enters the dissipation well can also dissipate energy. In the present study, the calculated section is shown in Figure1, where the section of volute chamber A-A at z = 1.9 m and the section of outlet tunnel B-B at Y + 1.8 m. The reference position of the potential energy is z = 0.0 m. The energy dissipation rate of the traditional energy dissipators is generally between 40% and 50%; any rate over 50% is considered excellent. In this paper, the energy dissipation rates exceeded 70% as shown in Table3, indicating that the vortex drop shaft without ventilation holes sufficiently dissipated energy and met the needs of the actual project. As shown in Figure 16, energy dissipation mainly occurred in the dissipation well and sloped section, and the dissipation rate of turbulent kinetic energy was generally between 20 and 30 m2·s−3 with a maximum of 32 m2·s−3. In addition, the wall friction dissipated some energy, and the dissipation rate of turbulent kinetic energy near the wall in the same cross-section was larger than that in other parts of the structure.

Table 3. The calculation results of the energy dissipation rate.

Section A-A Section B-B Condition Energy Dissipation Rate (%) V2/2g (m) H + p/fl (m) V2/2g (m) H + p/fl (m) Experiment 0.16 1.65 0.19 0.29 73.48 5% Simulation 0.17 1.69 0.20 0.28 74.12 Experiment 0.18 1.82 0.23 0.31 73.00 2% Simulation 0.19 1.81 0.24 0.33 71.66 Experiment 0.21 2.12 0.24 0.39 72.95 0.1% Water 2018,Simulation 10, x FOR PEER REVIEW 0.25 2.13 0.30 0.41 70.13 16 of 19 Notes: H is the elevation head, γ is the specific weight of water, p is the gauge pressure.

(a) Dissipation well (b) Section C-C (z = 0.14 m)

FigureFigure 16. 16.Dissipation Dissipation rate rate of of turbulent turbulent kinetic kinetic energy energy in in the the numerical numerical simulation simulation (P (P = 0.1%).= 0.1%). 5. Conclusions 5. Conclusion To address the negative pressure and cavitation erosion problems in vortex drop shaft spillways, To address the negative pressure and cavitation erosion problems in vortex drop shaft spillways, a new vortex drop shaft without ventilation holes is proposed in this paper. An increased height intake a new vortex drop shaft without ventilation holes is proposed in this paper. An increased height tunnel is used to facilitate aeration. The vortex drop shaft is simulated using the RNG k-ε turbulence intake tunnel is used to facilitate aeration. The vortex drop shaft is simulated using the RNG k-ε model. By comparing the experimental and simulation results, the hydraulic characteristics of the turbulence model. By comparing the experimental and simulation results, the hydraulic vortex drop shaft are studied in detail. The results of the research are as follows: characteristics of the vortex drop shaft are studied in detail. The results of the research are as follows: 1. The RNG k-ε turbulence model effectively simulated the flow characteristics of the vortex drop shaft. The hydraulic parameters, including the velocity and pressure, agreed well with the experimental data and showed the same trends. Thus, similar swirling problems can be simulated by the RNG k-ε turbulence model. 2. The flow regime in the vertical shaft was stable, but the flow in the dissipation well was swirling and turbulent. The energy dissipation rate exceeded 70% from the gradient section to the outlet tunnel, indicating the occurrence of sufficient energy dissipation. The velocity was small at the top and bottom but large in the middle, and the pressure was large at the top and bottom but small in the middle. Small negative pressure areas were observed. 3. By increasing the clearance height of the intake tunnel and changing the flow conditions, the water flowed into the vertical shaft with a large amount of air. There was a clear phenomenon of aeration and the aeration concentration could provide some reference for engineering design. Additionally, the cavitation number was larger than the critical value, which indicated a low probability of cavitation. However, our study has some limitations. For example, the diameters of the vertical shaft, vortex chamber and slope section need to be optimized based on existing body sizes to determine the best body shape for safe and stable operation. In addition, considering the scale effect, different scales of experiments and numerical simulations need to be performed.

Notations

2 𝐴 surface area (m ) 𝐶 aeration concentration

𝐶 coefficient of proportionality 𝜀 turbulent dissipation rate m2·s−3 𝐹 volume of fluid (VOF) function

𝐹 Froude number

−2 𝑓 gravity component (m·s ) Water 2018, 10, 1393 16 of 18

1. The RNG k-ε turbulence model effectively simulated the flow characteristics of the vortex drop shaft. The hydraulic parameters, including the velocity and pressure, agreed well with the experimental data and showed the same trends. Thus, similar swirling problems can be simulated by the RNG k-ε turbulence model. 2. The flow regime in the vertical shaft was stable, but the flow in the dissipation well was swirling and turbulent. The energy dissipation rate exceeded 70% from the gradient section to the outlet tunnel, indicating the occurrence of sufficient energy dissipation. The velocity was small at the top and bottom but large in the middle, and the pressure was large at the top and bottom but small in the middle. Small negative pressure areas were observed. 3. By increasing the clearance height of the intake tunnel and changing the flow conditions, the water flowed into the vertical shaft with a large amount of air. There was a clear phenomenon of aeration and the aeration concentration could provide some reference for engineering design. Additionally, the cavitation number was larger than the critical value, which indicated a low probability of cavitation.

However, our study has some limitations. For example, the diameters of the vertical shaft, vortex chamber and slope section need to be optimized based on existing body sizes to determine the best body shape for safe and stable operation. In addition, considering the scale effect, different scales of experiments and numerical simulations need to be performed.

Author Contributions: Conceptualization, W.Z. and J.W.; Methodology, W.Z.; Software, Z.D.; Validation, W.Z., J.W. and C.Z.; Formal Analysis, W.Z.; Investigation, W.Z.; Resources, J.W.; Data Curation, Z.Z.; Writing-Original Draft Preparation, W.Z.; Writing-Review & Editing, W.Z., Z.D. and Z.Z.; Supervision, J.W. Funding: This research received no external funding. Acknowledgments: The numerical calculations in this paper were performed using the supercomputing system at the Supercomputing Center of Wuhan University. Conﬂicts of Interest: The authors declare no conﬂict of interest.

Notations

2 As surface area (m ) C aeration concentration Cair coefficient of proportionality ε turbulent dissipation rate m2·s−3 F volume of fluid (VOF) function Fr Froude number −2 fi gravity component (m·s ) g gravity acceleration (m·s−2) −2 gn component of gravity normal to the free surface (m·s ) Gk generation of turbulent energy caused by the average velocity gradient h water depth (m) H elevation head (m) i slope k turbulent kinetic energy (kg·m2·s−2) LT turbulence length scale (m) N cavitation number P flood frequency −2 Pd disturbance energy per unit volume (N·m ) −2 Pt stabilising forces per unit volume (N·m ) p gauge pressure (Pa) p0 absolute pressure (Pa) pv vapor pressure (Pa) ρ density of water (kg·m−3) Water 2018, 10, 1393 17 of 18

Q discharge (L·s−1) r radius of the cavity (m) γ specific weight of water (N·m−3) R radius of vertical shaft (m) R0 hydraulic radius (m) S water trajectory (m) t time (s) ui velocity component (m/s) µ coefficient of dynamic viscosity (kg·m−1·s−1) −1 −1 µe f f revisionary coefficient of dynamic viscosity (kg·m ·s ) ν coefficient of kinematic viscosity (m2·s−1) V resultant velocity (m·s−1) −1 Vt tangential velocity (m·s ) −1 Vz vertical velocity (m·s ) δV volume of air entrained per unit time m3 σ coefficient of surface tension (N·m−1) θ angle between the velocity direction and the vertical direction λ factor of friction loss xi coordinate component

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